Theorem fusgrvtxfi 29780

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
fusgrvtxfi	⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isfusgr 29779	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3	2	simprbi 503	1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6533  Fincfn 8953  Vtxcvtx 29454  USGraphcusgr 29610  FinUSGraphcfusgr 29777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-fusgr 29778

This theorem is used by:  fusgrfupgrfs  29792  nbfusgrlevtxm1  29838  nbfusgrlevtxm2  29839  nbusgrvtxm1  29840  uvtxnm1nbgr  29865  cusgrm1rusgr  30043  wlksnfi  30376  fusgrhashclwwlkn  30550  clwwlkndivn  30551  fusgreghash2wsp  30819  numclwwlk3lem2  30865  numclwwlk4  30867